Concerning Puzzle 601 I'm sure others will notice all
those Fibonacci numbers in the examples given and see that:

It follows from the first Catalan identity (see,e.g., http://mathworld.wolfram.com/CatalansIdentity.html )
that if F(n) is the n-th Fibonacci number then for odd n we have F(n+2)
divides F(n)^2+1 and F(n) divides F(n+2)^2+1. So we can get examples by
finding n such that n is odd and both p = F(n) and q = F(n+2) are prime.
Examples up to 11000 were found for n = 3, 5, 11, 431, 569.

We shall show that there is an infinite number of integers, prime or
otherwise, that satisfy the conditions of the problem. It is well known
that the prime divisors d of integers of the form T^2 + 1 are d = 2 or d
= 4*v + 1 for some positive integer v. First we replace the condition
that p and q be primes with the conditions q > p and p and q are
relatively prime; e.g. (q, p) =1. Then there are integers k and l such
that

q^2 + 1 = p*k
(1)
and
p^2 + 1 = q*l
.
(2)

Since q > p, equation (1) implies k > (q^2)/p >
q (3) and

equation (2) implies pq > p^2 > ql or p > l
. (4)

Thus k > q > p >
l.
(5)

Then q^2 - p^2 = p*k - q*l or

q(q + l) = p(p +
k)
(6)

Hence (q + l )== 0 (mod p) and q^2 == l^2 (mod
p) (7)

Similarly (p + k ) == 0 (mod q) and p^2 == k^2 (mod
q) (8)

Thus q^2 + 1 == l^2 + 1 == 0 (mod
p) (9)

with l < p and we have descended to a new pair of integers (p, l)
such that l divides (p^2 + 1) and p divides (l^2 + 1).

We can continue this descent only a finite number of times until we
reach the representation 5 = 2^2 + 1^2. Then we can reverse the process
to ascend to an infinite number of pairs of relatively prime integers
(p, q) such that p divides (q^2 + 1) and q divides (p^2 + 1).

This is series No. A001519 on Sloane’s Online Encyclopedia of Integer
Sequences. Also starting with 2,5 13 ..., the terms of our series are
alternate terms of the Fibonacci series. Reimposing the requirement that
p and q be primes, we find the solutions (p, q) = (2, 5), (5, 13), (89,
233). After checking 100 terms of the sequence using UBASIC, no further
solutions were found.

In addition the following information was found on the internet. The
website

There is a complete list of all Fibonacci numbers and their factors
up to the 1000-th Fibonacci and 1000-th Lucas numbers and partial
results beyond that on Blair Kelly's Factorisation pages. The hyperlink
to Blair Kelly’s Factorisation pages goes to
http://mersennus.net/fibonacci/ which in turn refers to a table of
Fibonacci factorisations for n < 1,000 which is found at
http://mersennus.net/fibonacci/f1000.txt

Let n be odd in all the following. If r=2 then the identity becomes
F(n)^2-F(n+2)*F(n-2) = (-1)^(n-2)*F(2)^2 = -1*1^2 = -1.
So we have F(n)^2+1 = F(n+2)*F(n-2). This holds for all odd n,
so we also have F(n+2)^2+1 = F(n+4)*F(n).
Let (p, q) = (F(n), F(n+2)). Then q divides p^2+1, and p divides q^2+1.
If (p, q) are both prime then the puzzle conditions are satisfied.
If F(n) is prime for n<>4 then n also has to be prime, so (n, n+2) must
be
twin primes to give a chance. We get solutions for n = 3, 5, 11, 431,
569,
and probably no others according to heuristics.
n = 3, 5, 11 gives the three mentioned solutions (2, 5), (5, 13), (89,
233).
n = 431 gives (p, q) with 90 and 91 digits. n = 569 gives 119 digits in
each.

I don't know whether all solutions to the puzzle are of the Fibonacci
form.
A computer search only found such solutions for p < 10^9.
If we also allow solutions with composite p and/or q then a computer
search
still only found Fibonacci solutions for p < 10^8.

Since x and y have no divisor in common, it follows from the last
congruence that x^2+1 is divisible by y.
I.e.: (x,y) is also a 'solution'.

This makes it possible to construct a chain of solutions (a,b), (b,c),
... , (x,y),(y,z) with a as small as possible.
Hence, this leads us to a = 1, b = 2, c = 5, d = 13, e = 34, f = 89,g =
233, h = 610, etc... . It is clear that there
is thus only one sequence of numbers : 1, 2, 5, 13, ... ,a(n), ... such
that ( a(n-1) , a(n) ) is a solution of our
problem. It is not difficult to recognize the Fibonacci numbers
F(2n+1) as our a(n). Also, the recursive relation
a(n+1) = 3 a(n) - a(n-1) is easy to prove by induction. With Mathematica,
it was possible to find two 'prime solutions' :